binary search trees -...

Binary Search Trees

1 Sorting Numbers using a Treea sorting algorithmusing a tree of integer numbers

2 Header Filesdefining a node structdefining a tree class

3 Definition of Methodsselectors, methods insert() and to_string()the depth of a tree and a membership function

MCS 360 Lecture 24Introduction to Data StructuresJan Verschelde, 9 March 2020

Introduction to Data Structures (MCS 360) Binary Search Trees L-24 9 March 2020 1 / 28

Binary Search Trees





Sorting Numbers using a TreeConsider the sequence 4, 5, 2, 3, 8, 1, 7

Insert the numbers in a tree:

4PPP

5��

2HH

3

HH

8

��

1�

7

Rules to insert x at node N:if N is empty, then put x in Nif x < N, insert x to the left of Nif x ≥ N, insert x to the right of N

Recursive printing: left, node, right sorts the sequence.


running the program

We generate a sequence of random numbers.

$ /tmp/inttreeGive n : 8inserting 36inserting 471inserting 297inserting 453inserting 142inserting 917inserting 259inserting 123tree inorder string :36 123 142 259 297 453 471 917

$


Binary Search Trees





the main program#include "mcs360_integer_tree.h"using namespace mcs360_integer_tree;

int main(){

cout << "Give n : ";int n; cin >> n;srand(time(0));Tree T;for(int i=0; i<n; i++){

int r = rand() % 1000;cout << "inserting " << r << endl;T.insert(r);

}cout << "tree inorder string : " << endl;cout << T.to_string() << endl;return 0;

}


Binary Tree of Nodes

A binary tree consists of nodes, a node has a data field and twopointers, to left and right child.

Node

data

left s -

right s -

If both left and right point to NULL, then the node is a leaf.

The root of the tree is a pointer to a node.

In a binary search tree, for every node: all elements in the left subtreeare smaller than the data element in the node and larger elements arestored in the right subtree.


binary tree ADTabstract <typename T> binary_tree;/* A binary tree is either empty or it has a data element

and at most two subtrees, a left and a right subtree. */

abstract bool empty ( binary_tree t );postcondition: empty(t)

== true if t is empty,== false if t is not empty;

abstract T data_element ( binary_tree t );precondition: not empty(t);postcondition: data_element(t) is the data element of t;

abstract binary_tree left_subtree ( binary_tree t );precondition: not empty(t);postcondition: left_subtree(t) is the left subtree of t;

abstract binary_tree right_tree ( binary_tree t );precondition: not empty(t);postcondition: right_subtree(t) is the right subtree of t;


binary search tree ADT

A binary search tree stores items of a typefor which the comparison operation < is defined.

A binary search tree is a binary tree with an insert operation.

abstract <typename T> binary_search_tree;/* A binary search tree is a binary tree where everything

less than the data element is in the left subtreeand everyting else is in the right subtree. */

abstract void insert ( binary_search_tree t, T item );postcondition: if empty(t), then after insert(t, item)we have item = data_element(t), it not empty(t), then:if item < data_element(t), item is in left_subtree(t)if item >= data_element(t), item is in right_subtree(t);

abstract bool member ( binary_search_tree t, T item );postcondition: true if the item belongs to t,false otherwise.


Binary Search Trees





a node struct

The file mcs360_integer_tree_node.h contains

#ifndef __TREE_NODE_H__#define __TREE_NODE_H__#include <sstream>

struct Node{

int data; // numbers stored at node in treeNode *left; // pointer to left branch of treeNode *right; // pointer to right branch of tree

Node(const int& item, Node* left_ptr = NULL,Node* right_ptr = NULL) :

data(item),left(left_ptr), right(right_ptr) {}


virtual methods

virtual ~Node() {}

virtual std::string to_string() const{

std::ostringstream os;os << data;return os.str();

}};

#endif

By the virtual we can override later.


Binary Search Trees





defining a Tree class

The file mcs360_integer_tree.h contains

#ifndef __MCS360_INTEGER_TREE_H__#define __MCS360_INTEGER_TREE_H__#include "mcs360_integer_tree_node.h"#include <string>

namespace mcs360_integer_tree{

class Tree{

private:

Node *root; // data member

// construct tree from a nodeTree(Node *r) : root(r) {}


public methods

public:

Tree() : root(NULL) {}

Tree(const int& item,const Tree& left = Tree(),const Tree& right = Tree() ) :

root(new Node(item,left.root,right.root)) {}

Tree get_left() const;Tree get_right() const;

void insert(int item);std::string to_string();


Binary Search Trees





get_left() and get_right()

The file mcs360_integer_tree.cpp contains

#include "mcs360_integer_tree.h"

namespace mcs360_integer_tree{

Tree Tree::get_left() const{

return Tree(root->left);}

Tree Tree::get_right() const{

return Tree(root->right);}


inserting a number

void Tree::insert(int item){

if(root == NULL)root = new Node(item);

else if(item < root->data){

Tree L = this->get_left();L.insert(item);root->left = L.root;

}else{

Tree R = this->get_right();R.insert(item);root->right = R.root;

}}


writing the tree inorder

std::string Tree::to_string(){

using std::string;if(root == NULL)

return "";else{

string L = this->get_left().to_string();string d = root->to_string();string R = this->get_right().to_string();return L + " " + d + " " + R;

}}


tree traversals

Three kinds of tree traversals:

preorder: visit node, left tree, and right tree

inorder: visit left tree, node, and right tree

postorder: visit left tree, right tree, and node


Binary Search Trees





the Depth of a TreeThe depth of a node is zero is it is the root,otherwise it is 1 + the depth of its parent.

Need two new selector methods:bool Tree::is_left_null() const{

return (root->left == NULL);}bool Tree::is_right_null() const{

return (root->right == NULL);}

Need extra method to get data field:int Tree::get_data() const{

return root->data;}


a function depth()

int depth ( Tree t ){

int L = 0;if(!t.is_left_null())

L = 1 + depth(t.get_left());

int R = 0;if(!t.is_right_null())

R = 1 + depth(t.get_right());

return (L > R) ? L : R;}

Note: not a member of the class.


visualizing the depthInserting 9 numbers:

49 919 366 307 332 665 955 614 525tree inorder string :49 307 332 366 525 614 665 919 955

depth of tree : 549919366307332

665614525

955

Insert 2 spaces for every level.


write with depth

void write_with_depth ( int k, Tree t );// writes the tree t keeping track of the depth// with the parameter k, call initially with k = 0

void write_with_depth ( int k, Tree t ){

for(int i=0; i<k; i++) cout << " ";cout << t.get_data() << endl;if(!t.is_left_null())

write_with_depth(k+1,t.get_left());if(!t.is_right_null())

write_with_depth(k+1,t.get_right());}


navigating the tree

Does an integer number belong to the tree?

Recursive algorithm is_in(T,e):if e is in current node, return trueif e < data in current node,then return is_in(T.get_left(),e);else return is_in(T.get_right(),e).


the function is_in()bool is_in ( Tree t, int e ){

if(e == t.get_data())return true;

else if(e < t.get_data()){

if(t.is_left_null())return false;

elsereturn is_in(t.get_left(),e);

}else{

if(t.is_right_null())return false;

elsereturn is_in(t.get_right(),e);

}}


Summary + Exercises

We started with binary search trees covered in §8.4.

Exercises:1 Adjust the get_left() method to throw an exception in case the

node is NULL.2 Change the to_string() method so that the numbers are

printed in decreasing order.3 Modify mcs360_integer_tree.h and .cpp to define a

templated class for any numeric type.4 Use the code of this lecture to sort a vector of integer numbers.

Write a function that takes on input an STL vector<int> andthat returns the sorted vector.


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